LMFDB - Embedded newform 4006.2.a.f.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9880710497$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.232858 q^{3} +1.00000 q^{4} -1.02641 q^{5} +0.232858 q^{6} -0.709779 q^{7} +1.00000 q^{8} -2.94578 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.232858 q^{3} +1.00000 q^{4} -1.02641 q^{5} +0.232858 q^{6} -0.709779 q^{7} +1.00000 q^{8} -2.94578 q^{9} -1.02641 q^{10} +3.09122 q^{11} +0.232858 q^{12} -3.68227 q^{13} -0.709779 q^{14} -0.239008 q^{15} +1.00000 q^{16} +4.38444 q^{17} -2.94578 q^{18} -4.12034 q^{19} -1.02641 q^{20} -0.165277 q^{21} +3.09122 q^{22} +8.56854 q^{23} +0.232858 q^{24} -3.94648 q^{25} -3.68227 q^{26} -1.38452 q^{27} -0.709779 q^{28} -9.20965 q^{29} -0.239008 q^{30} +0.0180573 q^{31} +1.00000 q^{32} +0.719815 q^{33} +4.38444 q^{34} +0.728527 q^{35} -2.94578 q^{36} -10.1967 q^{37} -4.12034 q^{38} -0.857443 q^{39} -1.02641 q^{40} +3.04649 q^{41} -0.165277 q^{42} +1.55014 q^{43} +3.09122 q^{44} +3.02359 q^{45} +8.56854 q^{46} +1.74060 q^{47} +0.232858 q^{48} -6.49621 q^{49} -3.94648 q^{50} +1.02095 q^{51} -3.68227 q^{52} -2.32776 q^{53} -1.38452 q^{54} -3.17287 q^{55} -0.709779 q^{56} -0.959452 q^{57} -9.20965 q^{58} -3.33897 q^{59} -0.239008 q^{60} -4.58310 q^{61} +0.0180573 q^{62} +2.09085 q^{63} +1.00000 q^{64} +3.77953 q^{65} +0.719815 q^{66} +10.4253 q^{67} +4.38444 q^{68} +1.99525 q^{69} +0.728527 q^{70} -2.31151 q^{71} -2.94578 q^{72} -13.9467 q^{73} -10.1967 q^{74} -0.918966 q^{75} -4.12034 q^{76} -2.19409 q^{77} -0.857443 q^{78} -4.91632 q^{79} -1.02641 q^{80} +8.51494 q^{81} +3.04649 q^{82} -5.06288 q^{83} -0.165277 q^{84} -4.50025 q^{85} +1.55014 q^{86} -2.14454 q^{87} +3.09122 q^{88} -3.08382 q^{89} +3.02359 q^{90} +2.61360 q^{91} +8.56854 q^{92} +0.00420477 q^{93} +1.74060 q^{94} +4.22917 q^{95} +0.232858 q^{96} -14.6988 q^{97} -6.49621 q^{98} -9.10606 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) $ 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9} - 23 q^{10} - 32 q^{11} - 13 q^{12} - 8 q^{13} - 18 q^{14} - 14 q^{15} + 31 q^{16} - 30 q^{17} + 20 q^{18} - 38 q^{19} - 23 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 13 q^{24} + 40 q^{25} - 8 q^{26} - 28 q^{27} - 18 q^{28} - 7 q^{29} - 14 q^{30} - 32 q^{31} + 31 q^{32} - 9 q^{33} - 30 q^{34} - 14 q^{35} + 20 q^{36} + 11 q^{37} - 38 q^{38} - 9 q^{39} - 23 q^{40} - 76 q^{41} - 16 q^{42} - 33 q^{43} - 32 q^{44} - 40 q^{45} - 24 q^{46} - 96 q^{47} - 13 q^{48} + 15 q^{49} + 40 q^{50} - 55 q^{51} - 8 q^{52} - 28 q^{53} - 28 q^{54} - 52 q^{55} - 18 q^{56} - 21 q^{57} - 7 q^{58} - 72 q^{59} - 14 q^{60} - 9 q^{61} - 32 q^{62} - 54 q^{63} + 31 q^{64} - 38 q^{65} - 9 q^{66} - 4 q^{67} - 30 q^{68} - 17 q^{69} - 14 q^{70} - 61 q^{71} + 20 q^{72} - 62 q^{73} + 11 q^{74} - 63 q^{75} - 38 q^{76} - 9 q^{77} - 9 q^{78} - 30 q^{79} - 23 q^{80} - 13 q^{81} - 76 q^{82} - 90 q^{83} - 16 q^{84} + 26 q^{85} - 33 q^{86} - 34 q^{87} - 32 q^{88} - 99 q^{89} - 40 q^{90} - 47 q^{91} - 24 q^{92} - 6 q^{93} - 96 q^{94} - 24 q^{95} - 13 q^{96} - 46 q^{97} + 15 q^{98} - 48 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9 - 23 * q^10 - 32 * q^11 - 13 * q^12 - 8 * q^13 - 18 * q^14 - 14 * q^15 + 31 * q^16 - 30 * q^17 + 20 * q^18 - 38 * q^19 - 23 * q^20 - 16 * q^21 - 32 * q^22 - 24 * q^23 - 13 * q^24 + 40 * q^25 - 8 * q^26 - 28 * q^27 - 18 * q^28 - 7 * q^29 - 14 * q^30 - 32 * q^31 + 31 * q^32 - 9 * q^33 - 30 * q^34 - 14 * q^35 + 20 * q^36 + 11 * q^37 - 38 * q^38 - 9 * q^39 - 23 * q^40 - 76 * q^41 - 16 * q^42 - 33 * q^43 - 32 * q^44 - 40 * q^45 - 24 * q^46 - 96 * q^47 - 13 * q^48 + 15 * q^49 + 40 * q^50 - 55 * q^51 - 8 * q^52 - 28 * q^53 - 28 * q^54 - 52 * q^55 - 18 * q^56 - 21 * q^57 - 7 * q^58 - 72 * q^59 - 14 * q^60 - 9 * q^61 - 32 * q^62 - 54 * q^63 + 31 * q^64 - 38 * q^65 - 9 * q^66 - 4 * q^67 - 30 * q^68 - 17 * q^69 - 14 * q^70 - 61 * q^71 + 20 * q^72 - 62 * q^73 + 11 * q^74 - 63 * q^75 - 38 * q^76 - 9 * q^77 - 9 * q^78 - 30 * q^79 - 23 * q^80 - 13 * q^81 - 76 * q^82 - 90 * q^83 - 16 * q^84 + 26 * q^85 - 33 * q^86 - 34 * q^87 - 32 * q^88 - 99 * q^89 - 40 * q^90 - 47 * q^91 - 24 * q^92 - 6 * q^93 - 96 * q^94 - 24 * q^95 - 13 * q^96 - 46 * q^97 + 15 * q^98 - 48 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.232858	0.134440	0.0672202	−	0.997738i	$-0.478587\pi$
$3$	0.232858	0.134440	0.0672202	+	0.997738i	$0.478587\pi$
$4$	1.00000	0.500000
$5$	−1.02641	−0.459026	−0.229513	−	0.973306i	$-0.573713\pi$
$5$	−1.02641	−0.459026	−0.229513	+	0.973306i	$0.573713\pi$
$6$	0.232858	0.0950637
$7$	−0.709779	−0.268271	−0.134136	−	0.990963i	$-0.542826\pi$
$7$	−0.709779	−0.268271	−0.134136	+	0.990963i	$0.542826\pi$
$8$	1.00000	0.353553
$9$	−2.94578	−0.981926
$10$	−1.02641	−0.324580
$11$	3.09122	0.932039	0.466019	−	0.884774i	$-0.345688\pi$
$11$	3.09122	0.932039	0.466019	+	0.884774i	$0.345688\pi$
$12$	0.232858	0.0672202
$13$	−3.68227	−1.02128	−0.510638	−	0.859796i	$-0.670591\pi$
$13$	−3.68227	−1.02128	−0.510638	+	0.859796i	$0.670591\pi$
$14$	−0.709779	−0.189697
$15$	−0.239008	−0.0617116
$16$	1.00000	0.250000
$17$	4.38444	1.06338	0.531691	−	0.846938i	$-0.321557\pi$
$17$	4.38444	1.06338	0.531691	+	0.846938i	$0.321557\pi$
$18$	−2.94578	−0.694326
$19$	−4.12034	−0.945270	−0.472635	−	0.881258i	$-0.656697\pi$
$19$	−4.12034	−0.945270	−0.472635	+	0.881258i	$0.656697\pi$
$20$	−1.02641	−0.229513
$21$	−0.165277	−0.0360665
$22$	3.09122	0.659051
$23$	8.56854	1.78666	0.893332	−	0.449397i	$-0.148361\pi$
$23$	8.56854	1.78666	0.893332	+	0.449397i	$0.148361\pi$
$24$	0.232858	0.0475318
$25$	−3.94648	−0.789295
$26$	−3.68227	−0.722152
$27$	−1.38452	−0.266451
$28$	−0.709779	−0.134136
$29$	−9.20965	−1.71019	−0.855095	−	0.518472i	$-0.826501\pi$
$29$	−9.20965	−1.71019	−0.855095	+	0.518472i	$0.826501\pi$
$30$	−0.239008	−0.0436367
$31$	0.0180573	0.00324318	0.00162159	−	0.999999i	$-0.499484\pi$
$31$	0.0180573	0.00324318	0.00162159	+	0.999999i	$0.499484\pi$
$32$	1.00000	0.176777
$33$	0.719815	0.125304
$34$	4.38444	0.751925
$35$	0.728527	0.123144
$36$	−2.94578	−0.490963
$37$	−10.1967	−1.67633	−0.838164	−	0.545418i	$-0.816371\pi$
$37$	−10.1967	−1.67633	−0.838164	+	0.545418i	$0.816371\pi$
$38$	−4.12034	−0.668407
$39$	−0.857443	−0.137301
$40$	−1.02641	−0.162290
$41$	3.04649	0.475781	0.237891	−	0.971292i	$-0.423544\pi$
$41$	3.04649	0.475781	0.237891	+	0.971292i	$0.423544\pi$
$42$	−0.165277	−0.0255029
$43$	1.55014	0.236394	0.118197	−	0.992990i	$-0.462289\pi$
$43$	1.55014	0.236394	0.118197	+	0.992990i	$0.462289\pi$
$44$	3.09122	0.466019
$45$	3.02359	0.450729
$46$	8.56854	1.26336
$47$	1.74060	0.253892	0.126946	−	0.991910i	$-0.459482\pi$
$47$	1.74060	0.253892	0.126946	+	0.991910i	$0.459482\pi$
$48$	0.232858	0.0336101
$49$	−6.49621	−0.928030
$50$	−3.94648	−0.558116
$51$	1.02095	0.142962
$52$	−3.68227	−0.510638
$53$	−2.32776	−0.319743	−0.159871	−	0.987138i	$-0.551108\pi$
$53$	−2.32776	−0.319743	−0.159871	+	0.987138i	$0.551108\pi$
$54$	−1.38452	−0.188409
$55$	−3.17287	−0.427830
$56$	−0.709779	−0.0948483
$57$	−0.959452	−0.127082
$58$	−9.20965	−1.20929
$59$	−3.33897	−0.434697	−0.217349	−	0.976094i	$-0.569741\pi$
$59$	−3.33897	−0.434697	−0.217349	+	0.976094i	$0.569741\pi$
$60$	−0.239008	−0.0308558
$61$	−4.58310	−0.586806	−0.293403	−	0.955989i	$-0.594788\pi$
$61$	−4.58310	−0.586806	−0.293403	+	0.955989i	$0.594788\pi$
$62$	0.0180573	0.00229328
$63$	2.09085	0.263423
$64$	1.00000	0.125000
$65$	3.77953	0.468793
$66$	0.719815	0.0886030
$67$	10.4253	1.27366	0.636829	−	0.771005i	$-0.280246\pi$
$67$	10.4253	1.27366	0.636829	+	0.771005i	$0.280246\pi$
$68$	4.38444	0.531691
$69$	1.99525	0.240200
$70$	0.728527	0.0870756
$71$	−2.31151	−0.274326	−0.137163	−	0.990549i	$-0.543798\pi$
$71$	−2.31151	−0.274326	−0.137163	+	0.990549i	$0.543798\pi$
$72$	−2.94578	−0.347163
$73$	−13.9467	−1.63234	−0.816168	−	0.577815i	$-0.803906\pi$
$73$	−13.9467	−1.63234	−0.816168	+	0.577815i	$0.803906\pi$
$74$	−10.1967	−1.18534
$75$	−0.918966	−0.106113
$76$	−4.12034	−0.472635
$77$	−2.19409	−0.250039
$78$	−0.857443	−0.0970863
$79$	−4.91632	−0.553129	−0.276565	−	0.960995i	$-0.589196\pi$
$79$	−4.91632	−0.553129	−0.276565	+	0.960995i	$0.589196\pi$
$80$	−1.02641	−0.114757
$81$	8.51494	0.946104
$82$	3.04649	0.336428
$83$	−5.06288	−0.555723	−0.277861	−	0.960621i	$-0.589626\pi$
$83$	−5.06288	−0.555723	−0.277861	+	0.960621i	$0.589626\pi$
$84$	−0.165277	−0.0180332
$85$	−4.50025	−0.488120
$86$	1.55014	0.167156
$87$	−2.14454	−0.229918
$88$	3.09122	0.329525
$89$	−3.08382	−0.326884	−0.163442	−	0.986553i	$-0.552260\pi$
$89$	−3.08382	−0.326884	−0.163442	+	0.986553i	$0.552260\pi$
$90$	3.02359	0.318714
$91$	2.61360	0.273979
$92$	8.56854	0.893332
$93$	0.00420477	0.000436015 0
$94$	1.74060	0.179529
$95$	4.22917	0.433904
$96$	0.232858	0.0237659
$97$	−14.6988	−1.49244	−0.746221	−	0.665699i	$-0.768134\pi$
$97$	−14.6988	−1.49244	−0.746221	+	0.665699i	$0.768134\pi$
$98$	−6.49621	−0.656217
$99$	−9.10606	−0.915193
$100$	−3.94648	−0.394648
$101$	6.45408	0.642205	0.321103	−	0.947044i	$-0.395947\pi$
$101$	6.45408	0.642205	0.321103	+	0.947044i	$0.395947\pi$
$102$	1.02095	0.101089
$103$	−0.330649	−0.0325798	−0.0162899	−	0.999867i	$-0.505185\pi$
$103$	−0.330649	−0.0325798	−0.0162899	+	0.999867i	$0.505185\pi$
$104$	−3.68227	−0.361076
$105$	0.169643	0.0165555
$106$	−2.32776	−0.226092
$107$	6.99962	0.676679	0.338340	−	0.941024i	$-0.390135\pi$
$107$	6.99962	0.676679	0.338340	+	0.941024i	$0.390135\pi$
$108$	−1.38452	−0.133225
$109$	−8.77788	−0.840769	−0.420384	−	0.907346i	$-0.638105\pi$
$109$	−8.77788	−0.840769	−0.420384	+	0.907346i	$0.638105\pi$
$110$	−3.17287	−0.302522
$111$	−2.37438	−0.225366
$112$	−0.709779	−0.0670679
$113$	−19.4695	−1.83154	−0.915769	−	0.401706i	$-0.868417\pi$
$113$	−19.4695	−1.83154	−0.915769	+	0.401706i	$0.868417\pi$
$114$	−0.959452	−0.0898609
$115$	−8.79487	−0.820125
$116$	−9.20965	−0.855095
$117$	10.8471	1.00282
$118$	−3.33897	−0.307377
$119$	−3.11198	−0.285275
$120$	−0.239008	−0.0218184
$121$	−1.44434	−0.131304
$122$	−4.58310	−0.414934
$123$	0.709397	0.0639642
$124$	0.0180573	0.00162159
$125$	9.18278	0.821333
$126$	2.09085	0.186268
$127$	−2.40903	−0.213767	−0.106884	−	0.994272i	$-0.534087\pi$
$127$	−2.40903	−0.213767	−0.106884	+	0.994272i	$0.534087\pi$
$128$	1.00000	0.0883883
$129$	0.360961	0.0317809
$130$	3.77953	0.331486
$131$	3.09293	0.270230	0.135115	−	0.990830i	$-0.456860\pi$
$131$	3.09293	0.270230	0.135115	+	0.990830i	$0.456860\pi$
$132$	0.719815	0.0626518
$133$	2.92453	0.253589
$134$	10.4253	0.900613
$135$	1.42109	0.122308
$136$	4.38444	0.375963
$137$	−15.7824	−1.34838	−0.674191	−	0.738557i	$-0.735508\pi$
$137$	−15.7824	−1.34838	−0.674191	+	0.738557i	$0.735508\pi$
$138$	1.99525	0.169847
$139$	−8.65991	−0.734524	−0.367262	−	0.930118i	$-0.619705\pi$
$139$	−8.65991	−0.734524	−0.367262	+	0.930118i	$0.619705\pi$
$140$	0.728527	0.0615718
$141$	0.405311	0.0341334
$142$	−2.31151	−0.193978
$143$	−11.3827	−0.951869
$144$	−2.94578	−0.245481
$145$	9.45291	0.785021
$146$	−13.9467	−1.15424
$147$	−1.51269	−0.124765
$148$	−10.1967	−0.838164
$149$	23.7737	1.94762	0.973810	−	0.227364i	$-0.0730106\pi$
$149$	23.7737	1.94762	0.973810	+	0.227364i	$0.0730106\pi$
$150$	−0.918966	−0.0750333
$151$	−10.5318	−0.857070	−0.428535	−	0.903525i	$-0.640970\pi$
$151$	−10.5318	−0.857070	−0.428535	+	0.903525i	$0.640970\pi$
$152$	−4.12034	−0.334204
$153$	−12.9156	−1.04416
$154$	−2.19409	−0.176805
$155$	−0.0185342	−0.00148871
$156$	−0.857443	−0.0686504
$157$	3.61981	0.288892	0.144446	−	0.989513i	$-0.453860\pi$
$157$	3.61981	0.288892	0.144446	+	0.989513i	$0.453860\pi$
$158$	−4.91632	−0.391121
$159$	−0.542037	−0.0429863
$160$	−1.02641	−0.0811451
$161$	−6.08177	−0.479311
$162$	8.51494	0.668997
$163$	−21.2608	−1.66528	−0.832638	−	0.553818i	$-0.813170\pi$
$163$	−21.2608	−1.66528	−0.832638	+	0.553818i	$0.813170\pi$
$164$	3.04649	0.237891
$165$	−0.738827	−0.0575176
$166$	−5.06288	−0.392955
$167$	−14.3376	−1.10948	−0.554740	−	0.832024i	$-0.687182\pi$
$167$	−14.3376	−1.10948	−0.554740	+	0.832024i	$0.687182\pi$
$168$	−0.165277	−0.0127514
$169$	0.559076	0.0430058
$170$	−4.50025	−0.345153
$171$	12.1376	0.928185
$172$	1.55014	0.118197
$173$	14.0069	1.06492	0.532462	−	0.846454i	$-0.321267\pi$
$173$	14.0069	1.06492	0.532462	+	0.846454i	$0.321267\pi$
$174$	−2.14454	−0.162577
$175$	2.80113	0.211745
$176$	3.09122	0.233010
$177$	−0.777505	−0.0584409
$178$	−3.08382	−0.231142
$179$	16.0862	1.20234	0.601169	−	0.799122i	$-0.294702\pi$
$179$	16.0862	1.20234	0.601169	+	0.799122i	$0.294702\pi$
$180$	3.02359	0.225365
$181$	9.91944	0.737306	0.368653	−	0.929567i	$-0.379819\pi$
$181$	9.91944	0.737306	0.368653	+	0.929567i	$0.379819\pi$
$182$	2.61360	0.193733
$183$	−1.06721	−0.0788904
$184$	8.56854	0.631681
$185$	10.4660	0.769478
$186$	0.00420477	0.000308309 0
$187$	13.5533	0.991114
$188$	1.74060	0.126946
$189$	0.982703	0.0714811
$190$	4.22917	0.306816
$191$	20.8836	1.51108	0.755541	−	0.655101i	$-0.227374\pi$
$191$	20.8836	1.51108	0.755541	+	0.655101i	$0.227374\pi$
$192$	0.232858	0.0168050
$193$	−15.9073	−1.14503	−0.572517	−	0.819893i	$-0.694033\pi$
$193$	−15.9073	−1.14503	−0.572517	+	0.819893i	$0.694033\pi$
$194$	−14.6988	−1.05532
$195$	0.880091	0.0630246
$196$	−6.49621	−0.464015
$197$	14.8401	1.05732	0.528658	−	0.848835i	$-0.322696\pi$
$197$	14.8401	1.05732	0.528658	+	0.848835i	$0.322696\pi$
$198$	−9.10606	−0.647139
$199$	23.6524	1.67668	0.838338	−	0.545151i	$-0.183528\pi$
$199$	23.6524	1.67668	0.838338	+	0.545151i	$0.183528\pi$
$200$	−3.94648	−0.279058
$201$	2.42762	0.171231
$202$	6.45408	0.454108
$203$	6.53682	0.458795
$204$	1.02095	0.0714808
$205$	−3.12695	−0.218396
$206$	−0.330649	−0.0230374
$207$	−25.2410	−1.75437
$208$	−3.68227	−0.255319
$209$	−12.7369	−0.881029
$210$	0.169643	0.0117065
$211$	20.1077	1.38427	0.692136	−	0.721767i	$-0.256670\pi$
$211$	20.1077	1.38427	0.692136	+	0.721767i	$0.256670\pi$
$212$	−2.32776	−0.159871
$213$	−0.538253	−0.0368805
$214$	6.99962	0.478485
$215$	−1.59108	−0.108511
$216$	−1.38452	−0.0942046
$217$	−0.0128167	−0.000870053 0
$218$	−8.77788	−0.594513
$219$	−3.24759	−0.219452
$220$	−3.17287	−0.213915
$221$	−16.1447	−1.08601
$222$	−2.37438	−0.159358
$223$	−2.44667	−0.163841	−0.0819205	−	0.996639i	$-0.526105\pi$
$223$	−2.44667	−0.163841	−0.0819205	+	0.996639i	$0.526105\pi$
$224$	−0.709779	−0.0474241
$225$	11.6254	0.775029
$226$	−19.4695	−1.29509
$227$	9.32125	0.618673	0.309336	−	0.950953i	$-0.399893\pi$
$227$	9.32125	0.618673	0.309336	+	0.950953i	$0.399893\pi$
$228$	−0.959452	−0.0635412
$229$	−23.7376	−1.56863	−0.784314	−	0.620364i	$-0.786985\pi$
$229$	−23.7376	−1.56863	−0.784314	+	0.620364i	$0.786985\pi$
$230$	−8.79487	−0.579916
$231$	−0.510910	−0.0336154
$232$	−9.20965	−0.604643
$233$	2.92846	0.191850	0.0959249	−	0.995389i	$-0.469419\pi$
$233$	2.92846	0.191850	0.0959249	+	0.995389i	$0.469419\pi$
$234$	10.8471	0.709099
$235$	−1.78657	−0.116543
$236$	−3.33897	−0.217349
$237$	−1.14480	−0.0743629
$238$	−3.11198	−0.201720
$239$	3.43686	0.222312	0.111156	−	0.993803i	$-0.464545\pi$
$239$	3.43686	0.222312	0.111156	+	0.993803i	$0.464545\pi$
$240$	−0.239008	−0.0154279
$241$	−15.8815	−1.02302	−0.511509	−	0.859278i	$-0.670913\pi$
$241$	−15.8815	−1.02302	−0.511509	+	0.859278i	$0.670913\pi$
$242$	−1.44434	−0.0928456
$243$	6.13632	0.393645
$244$	−4.58310	−0.293403
$245$	6.66780	0.425990
$246$	0.709397	0.0452295
$247$	15.1722	0.965383
$248$	0.0180573	0.00114664
$249$	−1.17893	−0.0747116
$250$	9.18278	0.580770
$251$	10.1721	0.642057	0.321028	−	0.947070i	$-0.395972\pi$
$251$	10.1721	0.642057	0.321028	+	0.947070i	$0.395972\pi$
$252$	2.09085	0.131711
$253$	26.4873	1.66524
$254$	−2.40903	−0.151156
$255$	−1.04792	−0.0656231
$256$	1.00000	0.0625000
$257$	−9.09834	−0.567539	−0.283769	−	0.958893i	$-0.591585\pi$
$257$	−9.09834	−0.567539	−0.283769	+	0.958893i	$0.591585\pi$
$258$	0.360961	0.0224725
$259$	7.23741	0.449711
$260$	3.77953	0.234396
$261$	27.1296	1.67928
$262$	3.09293	0.191082
$263$	−0.376560	−0.0232197	−0.0116098	−	0.999933i	$-0.503696\pi$
$263$	−0.376560	−0.0232197	−0.0116098	+	0.999933i	$0.503696\pi$
$264$	0.719815	0.0443015
$265$	2.38925	0.146770
$266$	2.92453	0.179315
$267$	−0.718091	−0.0439465
$268$	10.4253	0.636829
$269$	−12.8914	−0.786005	−0.393002	−	0.919537i	$-0.628564\pi$
$269$	−12.8914	−0.786005	−0.393002	+	0.919537i	$0.628564\pi$
$270$	1.42109	0.0864847
$271$	−19.0056	−1.15451	−0.577254	−	0.816564i	$-0.695876\pi$
$271$	−19.0056	−1.15451	−0.577254	+	0.816564i	$0.695876\pi$
$272$	4.38444	0.265846
$273$	0.608595	0.0368339
$274$	−15.7824	−0.953450
$275$	−12.1994	−0.735654
$276$	1.99525	0.120100
$277$	−0.575188	−0.0345597	−0.0172798	−	0.999851i	$-0.505501\pi$
$277$	−0.575188	−0.0345597	−0.0172798	+	0.999851i	$0.505501\pi$
$278$	−8.65991	−0.519387
$279$	−0.0531927	−0.00318457
$280$	0.728527	0.0435378
$281$	13.4474	0.802203	0.401101	−	0.916034i	$-0.368628\pi$
$281$	13.4474	0.802203	0.401101	+	0.916034i	$0.368628\pi$
$282$	0.405311	0.0241359
$283$	−31.2326	−1.85659	−0.928293	−	0.371851i	$-0.878723\pi$
$283$	−31.2326	−1.85659	−0.928293	+	0.371851i	$0.878723\pi$
$284$	−2.31151	−0.137163
$285$	0.984794	0.0583342
$286$	−11.3827	−0.673073
$287$	−2.16233	−0.127638
$288$	−2.94578	−0.173582
$289$	2.22331	0.130783
$290$	9.45291	0.555094
$291$	−3.42274	−0.200644
$292$	−13.9467	−0.816168
$293$	−4.83890	−0.282692	−0.141346	−	0.989960i	$-0.545143\pi$
$293$	−4.83890	−0.282692	−0.141346	+	0.989960i	$0.545143\pi$
$294$	−1.51269	−0.0882220
$295$	3.42717	0.199537
$296$	−10.1967	−0.592672
$297$	−4.27986	−0.248342
$298$	23.7737	1.37718
$299$	−31.5516	−1.82468
$300$	−0.918966	−0.0530566
$301$	−1.10026	−0.0634177
$302$	−10.5318	−0.606040
$303$	1.50288	0.0863383
$304$	−4.12034	−0.236318
$305$	4.70415	0.269359
$306$	−12.9156	−0.738335
$307$	4.14418	0.236521	0.118260	−	0.992983i	$-0.462268\pi$
$307$	4.14418	0.236521	0.118260	+	0.992983i	$0.462268\pi$
$308$	−2.19409	−0.125020
$309$	−0.0769941	−0.00438004
$310$	−0.0185342	−0.00105267
$311$	24.6227	1.39623	0.698114	−	0.715987i	$-0.254023\pi$
$311$	24.6227	1.39623	0.698114	+	0.715987i	$0.254023\pi$
$312$	−0.857443	−0.0485432
$313$	28.7216	1.62344	0.811721	−	0.584046i	$-0.198531\pi$
$313$	28.7216	1.62344	0.811721	+	0.584046i	$0.198531\pi$
$314$	3.61981	0.204278
$315$	−2.14608	−0.120918
$316$	−4.91632	−0.276565
$317$	20.6363	1.15905	0.579524	−	0.814955i	$-0.303239\pi$
$317$	20.6363	1.15905	0.579524	+	0.814955i	$0.303239\pi$
$318$	−0.542037	−0.0303959
$319$	−28.4691	−1.59396
$320$	−1.02641	−0.0573783
$321$	1.62992	0.0909730
$322$	−6.08177	−0.338924
$323$	−18.0654	−1.00518
$324$	8.51494	0.473052
$325$	14.5320	0.806089
$326$	−21.2608	−1.17753
$327$	−2.04400	−0.113033
$328$	3.04649	0.168214
$329$	−1.23544	−0.0681121
$330$	−0.738827	−0.0406711
$331$	24.8507	1.36592	0.682959	−	0.730456i	$-0.260693\pi$
$331$	24.8507	1.36592	0.682959	+	0.730456i	$0.260693\pi$
$332$	−5.06288	−0.277861
$333$	30.0372	1.64603
$334$	−14.3376	−0.784521
$335$	−10.7007	−0.584643
$336$	−0.165277	−0.00901662
$337$	−23.4110	−1.27528	−0.637639	−	0.770336i	$-0.720089\pi$
$337$	−23.4110	−1.27528	−0.637639	+	0.770336i	$0.720089\pi$
$338$	0.559076	0.0304097
$339$	−4.53362	−0.246233
$340$	−4.50025	−0.244060
$341$	0.0558191	0.00302277
$342$	12.1376	0.656326
$343$	9.57933	0.517235
$344$	1.55014	0.0835779
$345$	−2.04795	−0.110258
$346$	14.0069	0.753015
$347$	11.9997	0.644178	0.322089	−	0.946709i	$-0.395615\pi$
$347$	11.9997	0.644178	0.322089	+	0.946709i	$0.395615\pi$
$348$	−2.14454	−0.114959
$349$	19.1804	1.02670	0.513352	−	0.858178i	$-0.328403\pi$
$349$	19.1804	1.02670	0.513352	+	0.858178i	$0.328403\pi$
$350$	2.80113	0.149727
$351$	5.09817	0.272120
$352$	3.09122	0.164763
$353$	−20.7006	−1.10178	−0.550891	−	0.834577i	$-0.685712\pi$
$353$	−20.7006	−1.10178	−0.550891	+	0.834577i	$0.685712\pi$
$354$	−0.777505	−0.0413239
$355$	2.37257	0.125923
$356$	−3.08382	−0.163442
$357$	−0.724649	−0.0383525
$358$	16.0862	0.850181
$359$	18.0887	0.954685	0.477343	−	0.878717i	$-0.341600\pi$
$359$	18.0887	0.954685	0.477343	+	0.878717i	$0.341600\pi$
$360$	3.02359	0.159357
$361$	−2.02281	−0.106464
$362$	9.91944	0.521354
$363$	−0.336325	−0.0176525
$364$	2.61360	0.136990
$365$	14.3151	0.749285
$366$	−1.06721	−0.0557839
$367$	−29.5023	−1.54001	−0.770004	−	0.638039i	$-0.779746\pi$
$367$	−29.5023	−1.54001	−0.770004	+	0.638039i	$0.779746\pi$
$368$	8.56854	0.446666
$369$	−8.97427	−0.467182
$370$	10.4660	0.544103
$371$	1.65220	0.0857779
$372$	0.00420477	0.000218007 0
$373$	9.29628	0.481343	0.240672	−	0.970607i	$-0.422632\pi$
$373$	9.29628	0.481343	0.240672	+	0.970607i	$0.422632\pi$
$374$	13.5533	0.700823
$375$	2.13828	0.110420
$376$	1.74060	0.0897645
$377$	33.9124	1.74658
$378$	0.982703	0.0505448
$379$	21.4937	1.10406	0.552029	−	0.833825i	$-0.313854\pi$
$379$	21.4937	1.10406	0.552029	+	0.833825i	$0.313854\pi$
$380$	4.22917	0.216952
$381$	−0.560962	−0.0287389
$382$	20.8836	1.06850
$383$	9.28050	0.474212	0.237106	−	0.971484i	$-0.423801\pi$
$383$	9.28050	0.474212	0.237106	+	0.971484i	$0.423801\pi$
$384$	0.232858	0.0118830
$385$	2.25204	0.114775
$386$	−15.9073	−0.809661
$387$	−4.56636	−0.232121
$388$	−14.6988	−0.746221
$389$	−18.7534	−0.950834	−0.475417	−	0.879761i	$-0.657703\pi$
$389$	−18.7534	−0.950834	−0.475417	+	0.879761i	$0.657703\pi$
$390$	0.880091	0.0445651
$391$	37.5682	1.89991
$392$	−6.49621	−0.328108
$393$	0.720212	0.0363299
$394$	14.8401	0.747635
$395$	5.04618	0.253901
$396$	−9.10606	−0.457596
$397$	16.3237	0.819264	0.409632	−	0.912251i	$-0.365657\pi$
$397$	16.3237	0.819264	0.409632	+	0.912251i	$0.365657\pi$
$398$	23.6524	1.18559
$399$	0.680999	0.0340926
$400$	−3.94648	−0.197324
$401$	−6.39449	−0.319326	−0.159663	−	0.987172i	$-0.551041\pi$
$401$	−6.39449	−0.319326	−0.159663	+	0.987172i	$0.551041\pi$
$402$	2.42762	0.121079
$403$	−0.0664917	−0.00331219
$404$	6.45408	0.321103
$405$	−8.73984	−0.434286
$406$	6.53682	0.324417
$407$	−31.5203	−1.56240
$408$	1.02095	0.0505445
$409$	−22.7172	−1.12329	−0.561647	−	0.827377i	$-0.689832\pi$
$409$	−22.7172	−1.12329	−0.561647	+	0.827377i	$0.689832\pi$
$410$	−3.12695	−0.154429
$411$	−3.67505	−0.181277
$412$	−0.330649	−0.0162899
$413$	2.36993	0.116617
$414$	−25.2410	−1.24053
$415$	5.19660	0.255091
$416$	−3.68227	−0.180538
$417$	−2.01652	−0.0987496
$418$	−12.7369	−0.622981
$419$	20.3370	0.993528	0.496764	−	0.867886i	$-0.334521\pi$
$419$	20.3370	0.993528	0.496764	+	0.867886i	$0.334521\pi$
$420$	0.169643	0.00827773
$421$	12.5555	0.611916	0.305958	−	0.952045i	$-0.401023\pi$
$421$	12.5555	0.611916	0.305958	+	0.952045i	$0.401023\pi$
$422$	20.1077	0.978828
$423$	−5.12742	−0.249303
$424$	−2.32776	−0.113046
$425$	−17.3031	−0.839323
$426$	−0.538253	−0.0260784
$427$	3.25299	0.157423
$428$	6.99962	0.338340
$429$	−2.65055	−0.127970
$430$	−1.59108	−0.0767288
$431$	−17.1286	−0.825055	−0.412527	−	0.910945i	$-0.635354\pi$
$431$	−17.1286	−0.825055	−0.412527	+	0.910945i	$0.635354\pi$
$432$	−1.38452	−0.0666127
$433$	−31.0475	−1.49205	−0.746023	−	0.665920i	$-0.768039\pi$
$433$	−31.0475	−1.49205	−0.746023	+	0.665920i	$0.768039\pi$
$434$	−0.0128167	−0.000615221 0
$435$	2.20118	0.105539
$436$	−8.77788	−0.420384
$437$	−35.3053	−1.68888
$438$	−3.24759	−0.155176
$439$	27.5236	1.31363	0.656816	−	0.754051i	$-0.271903\pi$
$439$	27.5236	1.31363	0.656816	+	0.754051i	$0.271903\pi$
$440$	−3.17287	−0.151261
$441$	19.1364	0.911257
$442$	−16.1447	−0.767923
$443$	−32.4503	−1.54176	−0.770879	−	0.636981i	$-0.780183\pi$
$443$	−32.4503	−1.54176	−0.770879	+	0.636981i	$0.780183\pi$
$444$	−2.37438	−0.112683
$445$	3.16528	0.150048
$446$	−2.44667	−0.115853
$447$	5.53589	0.261839
$448$	−0.709779	−0.0335339
$449$	32.0691	1.51344	0.756718	−	0.653741i	$-0.226801\pi$
$449$	32.0691	1.51344	0.756718	+	0.653741i	$0.226801\pi$
$450$	11.6254	0.548028
$451$	9.41737	0.443446
$452$	−19.4695	−0.915769
$453$	−2.45242	−0.115225
$454$	9.32125	0.437468
$455$	−2.68263	−0.125764
$456$	−0.959452	−0.0449304
$457$	6.27608	0.293583	0.146791	−	0.989167i	$-0.453105\pi$
$457$	6.27608	0.293583	0.146791	+	0.989167i	$0.453105\pi$
$458$	−23.7376	−1.10919
$459$	−6.07034	−0.283339
$460$	−8.79487	−0.410063
$461$	−33.7409	−1.57147	−0.785734	−	0.618564i	$-0.787715\pi$
$461$	−33.7409	−1.57147	−0.785734	+	0.618564i	$0.787715\pi$
$462$	−0.510910	−0.0237697
$463$	22.6032	1.05046	0.525230	−	0.850961i	$-0.323979\pi$
$463$	22.6032	1.05046	0.525230	+	0.850961i	$0.323979\pi$
$464$	−9.20965	−0.427547
$465$	−0.00431584	−0.000200142 0
$466$	2.92846	0.135658
$467$	33.0095	1.52750	0.763749	−	0.645514i	$-0.223357\pi$
$467$	33.0095	1.52750	0.763749	+	0.645514i	$0.223357\pi$
$468$	10.8471	0.501409
$469$	−7.39970	−0.341686
$470$	−1.78657	−0.0824085
$471$	0.842900	0.0388388
$472$	−3.33897	−0.153689
$473$	4.79182	0.220328
$474$	−1.14480	−0.0525825
$475$	16.2608	0.746097
$476$	−3.11198	−0.142638
$477$	6.85707	0.313964
$478$	3.43686	0.157198
$479$	−29.5838	−1.35172	−0.675859	−	0.737031i	$-0.736227\pi$
$479$	−29.5838	−1.35172	−0.675859	+	0.737031i	$0.736227\pi$
$480$	−0.239008	−0.0109092
$481$	37.5470	1.71199
$482$	−15.8815	−0.723383
$483$	−1.41619	−0.0644387
$484$	−1.44434	−0.0656518
$485$	15.0871	0.685069
$486$	6.13632	0.278349
$487$	−5.88070	−0.266480	−0.133240	−	0.991084i	$-0.542538\pi$
$487$	−5.88070	−0.266480	−0.133240	+	0.991084i	$0.542538\pi$
$488$	−4.58310	−0.207467
$489$	−4.95074	−0.223880
$490$	6.66780	0.301220
$491$	−21.1468	−0.954342	−0.477171	−	0.878810i	$-0.658338\pi$
$491$	−21.1468	−0.954342	−0.477171	+	0.878810i	$0.658338\pi$
$492$	0.709397	0.0319821
$493$	−40.3792	−1.81859
$494$	15.1722	0.682629
$495$	9.34658	0.420097
$496$	0.0180573	0.000810796 0
$497$	1.64066	0.0735938
$498$	−1.17893	−0.0528291
$499$	−13.9679	−0.625287	−0.312643	−	0.949871i	$-0.601214\pi$
$499$	−13.9679	−0.625287	−0.312643	+	0.949871i	$0.601214\pi$
$500$	9.18278	0.410666
$501$	−3.33863	−0.149159
$502$	10.1721	0.454003
$503$	20.3580	0.907720	0.453860	−	0.891073i	$-0.350047\pi$
$503$	20.3580	0.907720	0.453860	+	0.891073i	$0.350047\pi$
$504$	2.09085	0.0931340
$505$	−6.62455	−0.294789
$506$	26.4873	1.17750
$507$	0.130185	0.00578172
$508$	−2.40903	−0.106884
$509$	−39.3557	−1.74441	−0.872205	−	0.489140i	$-0.837311\pi$
$509$	−39.3557	−1.74441	−0.872205	+	0.489140i	$0.837311\pi$
$510$	−1.04792	−0.0464025
$511$	9.89907	0.437909
$512$	1.00000	0.0441942
$513$	5.70469	0.251868
$514$	−9.09834	−0.401311
$515$	0.339382	0.0149550
$516$	0.360961	0.0158904
$517$	5.38058	0.236638
$518$	7.23741	0.317994
$519$	3.26161	0.143169
$520$	3.77953	0.165743
$521$	20.8645	0.914090	0.457045	−	0.889444i	$-0.348908\pi$
$521$	20.8645	0.914090	0.457045	+	0.889444i	$0.348908\pi$
$522$	27.1296	1.18743
$523$	−29.1729	−1.27564	−0.637821	−	0.770185i	$-0.720164\pi$
$523$	−29.1729	−1.27564	−0.637821	+	0.770185i	$0.720164\pi$
$524$	3.09293	0.135115
$525$	0.652263	0.0284671
$526$	−0.376560	−0.0164188
$527$	0.0791710	0.00344874
$528$	0.719815	0.0313259
$529$	50.4199	2.19217
$530$	2.38925	0.103782
$531$	9.83587	0.426841
$532$	2.92453	0.126795
$533$	−11.2180	−0.485904
$534$	−0.718091	−0.0310748
$535$	−7.18451	−0.310613
$536$	10.4253	0.450306
$537$	3.74579	0.161643
$538$	−12.8914	−0.555789
$539$	−20.0812	−0.864960
$540$	1.42109	0.0611539
$541$	13.2521	0.569754	0.284877	−	0.958564i	$-0.408047\pi$
$541$	13.2521	0.569754	0.284877	+	0.958564i	$0.408047\pi$
$542$	−19.0056	−0.816361
$543$	2.30982	0.0991237
$544$	4.38444	0.187981
$545$	9.00974	0.385935
$546$	0.608595	0.0260455
$547$	9.75364	0.417035	0.208518	−	0.978019i	$-0.433136\pi$
$547$	9.75364	0.417035	0.208518	+	0.978019i	$0.433136\pi$
$548$	−15.7824	−0.674191
$549$	13.5008	0.576200
$550$	−12.1994	−0.520186
$551$	37.9469	1.61659
$552$	1.99525	0.0849234
$553$	3.48950	0.148389
$554$	−0.575188	−0.0244374
$555$	2.43709	0.103449
$556$	−8.65991	−0.367262
$557$	36.8108	1.55972	0.779861	−	0.625952i	$-0.215290\pi$
$557$	36.8108	1.55972	0.779861	+	0.625952i	$0.215290\pi$
$558$	−0.0531927	−0.00225183
$559$	−5.70802	−0.241424
$560$	0.728527	0.0307859
$561$	3.15598	0.133246
$562$	13.4474	0.567243
$563$	34.8676	1.46950	0.734748	−	0.678341i	$-0.237301\pi$
$563$	34.8676	1.46950	0.734748	+	0.678341i	$0.237301\pi$
$564$	0.405311	0.0170667
$565$	19.9838	0.840723
$566$	−31.2326	−1.31280
$567$	−6.04373	−0.253813
$568$	−2.31151	−0.0969888
$569$	−8.43012	−0.353409	−0.176704	−	0.984264i	$-0.556544\pi$
$569$	−8.43012	−0.353409	−0.176704	+	0.984264i	$0.556544\pi$
$570$	0.984794	0.0412485
$571$	25.3349	1.06023	0.530116	−	0.847925i	$-0.322148\pi$
$571$	25.3349	1.06023	0.530116	+	0.847925i	$0.322148\pi$
$572$	−11.3827	−0.475935
$573$	4.86290	0.203150
$574$	−2.16233	−0.0902540
$575$	−33.8155	−1.41021
$576$	−2.94578	−0.122741
$577$	30.4464	1.26750	0.633750	−	0.773538i	$-0.281515\pi$
$577$	30.4464	1.26750	0.633750	+	0.773538i	$0.281515\pi$
$578$	2.22331	0.0924773
$579$	−3.70414	−0.153939
$580$	9.45291	0.392511
$581$	3.59353	0.149085
$582$	−3.42274	−0.141877
$583$	−7.19564	−0.298013
$584$	−13.9467	−0.577118
$585$	−11.1336	−0.460319
$586$	−4.83890	−0.199893
$587$	−5.19103	−0.214257	−0.107128	−	0.994245i	$-0.534166\pi$
$587$	−5.19103	−0.214257	−0.107128	+	0.994245i	$0.534166\pi$
$588$	−1.51269	−0.0623824
$589$	−0.0744021	−0.00306569
$590$	3.42717	0.141094
$591$	3.45563	0.142146
$592$	−10.1967	−0.419082
$593$	−24.3895	−1.00156	−0.500779	−	0.865575i	$-0.666953\pi$
$593$	−24.3895	−1.00156	−0.500779	+	0.865575i	$0.666953\pi$
$594$	−4.27986	−0.175605
$595$	3.19418	0.130949
$596$	23.7737	0.973810
$597$	5.50765	0.225413
$598$	−31.5516	−1.29024
$599$	−24.8087	−1.01366	−0.506828	−	0.862047i	$-0.669182\pi$
$599$	−24.8087	−1.01366	−0.506828	+	0.862047i	$0.669182\pi$
$600$	−0.918966	−0.0375166
$601$	35.4386	1.44557	0.722786	−	0.691072i	$-0.242861\pi$
$601$	35.4386	1.44557	0.722786	+	0.691072i	$0.242861\pi$
$602$	−1.10026	−0.0448431
$603$	−30.7108	−1.25064
$604$	−10.5318	−0.428535
$605$	1.48249	0.0602718
$606$	1.50288	0.0610504
$607$	5.56232	0.225768	0.112884	−	0.993608i	$-0.463991\pi$
$607$	5.56232	0.225768	0.112884	+	0.993608i	$0.463991\pi$
$608$	−4.12034	−0.167102
$609$	1.52215	0.0616805
$610$	4.70415	0.190466
$611$	−6.40934	−0.259294
$612$	−12.9156	−0.522081
$613$	4.67101	0.188660	0.0943302	−	0.995541i	$-0.469929\pi$
$613$	4.67101	0.188660	0.0943302	+	0.995541i	$0.469929\pi$
$614$	4.14418	0.167245
$615$	−0.728135	−0.0293612
$616$	−2.19409	−0.0884023
$617$	−32.2861	−1.29979	−0.649895	−	0.760024i	$-0.725187\pi$
$617$	−32.2861	−1.29979	−0.649895	+	0.760024i	$0.725187\pi$
$618$	−0.0769941	−0.00309716
$619$	−17.7036	−0.711567	−0.355783	−	0.934568i	$-0.615786\pi$
$619$	−17.7036	−0.711567	−0.355783	+	0.934568i	$0.615786\pi$
$620$	−0.0185342	−0.000744353 0
$621$	−11.8633	−0.476058
$622$	24.6227	0.987282
$623$	2.18883	0.0876937
$624$	−0.857443	−0.0343252
$625$	10.3070	0.412282
$626$	28.7216	1.14795
$627$	−2.96588	−0.118446
$628$	3.61981	0.144446
$629$	−44.7068	−1.78258
$630$	−2.14608	−0.0855018
$631$	−10.4197	−0.414801	−0.207400	−	0.978256i	$-0.566500\pi$
$631$	−10.4197	−0.414801	−0.207400	+	0.978256i	$0.566500\pi$
$632$	−4.91632	−0.195561
$633$	4.68223	0.186102
$634$	20.6363	0.819570
$635$	2.47266	0.0981247
$636$	−0.542037	−0.0214932
$637$	23.9208	0.947776
$638$	−28.4691	−1.12710
$639$	6.80920	0.269368
$640$	−1.02641	−0.0405726
$641$	−7.50493	−0.296427	−0.148213	−	0.988955i	$-0.547352\pi$
$641$	−7.50493	−0.296427	−0.148213	+	0.988955i	$0.547352\pi$
$642$	1.62992	0.0643276
$643$	33.5801	1.32427	0.662135	−	0.749385i	$-0.269651\pi$
$643$	33.5801	1.32427	0.662135	+	0.749385i	$0.269651\pi$
$644$	−6.08177	−0.239655
$645$	−0.370496	−0.0145882
$646$	−18.0654	−0.710773
$647$	−31.0440	−1.22046	−0.610232	−	0.792223i	$-0.708924\pi$
$647$	−31.0440	−1.22046	−0.610232	+	0.792223i	$0.708924\pi$
$648$	8.51494	0.334498
$649$	−10.3215	−0.405155
$650$	14.5320	0.569991
$651$	−0.00298446	−0.000116970 0
$652$	−21.2608	−0.832638
$653$	−3.73585	−0.146195	−0.0730975	−	0.997325i	$-0.523288\pi$
$653$	−3.73585	−0.146195	−0.0730975	+	0.997325i	$0.523288\pi$
$654$	−2.04400	−0.0799266
$655$	−3.17462	−0.124043
$656$	3.04649	0.118945
$657$	41.0838	1.60283
$658$	−1.23544	−0.0481625
$659$	47.1290	1.83588	0.917942	−	0.396714i	$-0.129849\pi$
$659$	47.1290	1.83588	0.917942	+	0.396714i	$0.129849\pi$
$660$	−0.738827	−0.0287588
$661$	5.60470	0.217998	0.108999	−	0.994042i	$-0.465236\pi$
$661$	5.60470	0.217998	0.108999	+	0.994042i	$0.465236\pi$
$662$	24.8507	0.965850
$663$	−3.75941	−0.146003
$664$	−5.06288	−0.196478
$665$	−3.00178	−0.116404
$666$	30.0372	1.16392
$667$	−78.9133	−3.05553
$668$	−14.3376	−0.554740
$669$	−0.569725	−0.0220268
$670$	−10.7007	−0.413405
$671$	−14.1674	−0.546926
$672$	−0.165277	−0.00637572
$673$	2.45163	0.0945034	0.0472517	−	0.998883i	$-0.484954\pi$
$673$	2.45163	0.0945034	0.0472517	+	0.998883i	$0.484954\pi$
$674$	−23.4110	−0.901757
$675$	5.46397	0.210308
$676$	0.559076	0.0215029
$677$	13.9193	0.534962	0.267481	−	0.963563i	$-0.413809\pi$
$677$	13.9193	0.534962	0.267481	+	0.963563i	$0.413809\pi$
$678$	−4.53362	−0.174113
$679$	10.4329	0.400379
$680$	−4.50025	−0.172577
$681$	2.17052	0.0831746
$682$	0.0558191	0.00213742
$683$	29.8592	1.14253	0.571265	−	0.820766i	$-0.306453\pi$
$683$	29.8592	1.14253	0.571265	+	0.820766i	$0.306453\pi$
$684$	12.1376	0.464093
$685$	16.1993	0.618942
$686$	9.57933	0.365741
$687$	−5.52749	−0.210887
$688$	1.55014	0.0590985
$689$	8.57144	0.326546
$690$	−2.04795	−0.0779641
$691$	−48.0239	−1.82691	−0.913457	−	0.406935i	$-0.866597\pi$
$691$	−48.0239	−1.82691	−0.913457	+	0.406935i	$0.866597\pi$
$692$	14.0069	0.532462
$693$	6.46329	0.245520
$694$	11.9997	0.455502
$695$	8.88864	0.337165
$696$	−2.14454	−0.0812884
$697$	13.3571	0.505937
$698$	19.1804	0.725990
$699$	0.681914	0.0257924
$700$	2.80113	0.105873
$701$	−43.5263	−1.64396	−0.821982	−	0.569513i	$-0.807132\pi$
$701$	−43.5263	−1.64396	−0.821982	+	0.569513i	$0.807132\pi$
$702$	5.09817	0.192418
$703$	42.0139	1.58458
$704$	3.09122	0.116505
$705$	−0.416017	−0.0156681
$706$	−20.7006	−0.779078
$707$	−4.58097	−0.172285
$708$	−0.777505	−0.0292204
$709$	−6.89981	−0.259128	−0.129564	−	0.991571i	$-0.541358\pi$
$709$	−6.89981	−0.259128	−0.129564	+	0.991571i	$0.541358\pi$
$710$	2.37257	0.0890408
$711$	14.4824	0.543132
$712$	−3.08382	−0.115571
$713$	0.154725	0.00579448
$714$	−0.724649	−0.0271193
$715$	11.6834	0.436933
$716$	16.0862	0.601169
$717$	0.800299	0.0298877
$718$	18.0887	0.675064
$719$	−20.1399	−0.751091	−0.375546	−	0.926804i	$-0.622545\pi$
$719$	−20.1399	−0.751091	−0.375546	+	0.926804i	$0.622545\pi$
$720$	3.02359	0.112682
$721$	0.234688	0.00874023
$722$	−2.02281	−0.0752813
$723$	−3.69813	−0.137535
$724$	9.91944	0.368653
$725$	36.3457	1.34984
$726$	−0.336325	−0.0124822
$727$	−9.17064	−0.340120	−0.170060	−	0.985434i	$-0.554396\pi$
$727$	−9.17064	−0.340120	−0.170060	+	0.985434i	$0.554396\pi$
$728$	2.61360	0.0968663
$729$	−24.1159	−0.893182
$730$	14.3151	0.529824
$731$	6.79649	0.251377
$732$	−1.06721	−0.0394452
$733$	27.0583	0.999420	0.499710	−	0.866193i	$-0.333440\pi$
$733$	27.0583	0.999420	0.499710	+	0.866193i	$0.333440\pi$
$734$	−29.5023	−1.08895
$735$	1.55265	0.0572703
$736$	8.56854	0.315841
$737$	32.2271	1.18710
$738$	−8.97427	−0.330347
$739$	29.7275	1.09354	0.546772	−	0.837282i	$-0.315857\pi$
$739$	29.7275	1.09354	0.546772	+	0.837282i	$0.315857\pi$
$740$	10.4660	0.384739
$741$	3.53296	0.129786
$742$	1.65220	0.0606541
$743$	−20.7854	−0.762542	−0.381271	−	0.924463i	$-0.624513\pi$
$743$	−20.7854	−0.762542	−0.381271	+	0.924463i	$0.624513\pi$
$744$	0.00420477	0.000154154 0
$745$	−24.4017	−0.894008
$746$	9.29628	0.340361
$747$	14.9141	0.545679
$748$	13.5533	0.495557
$749$	−4.96819	−0.181534
$750$	2.13828	0.0780789
$751$	17.5388	0.640001	0.320001	−	0.947417i	$-0.396317\pi$
$751$	17.5388	0.640001	0.320001	+	0.947417i	$0.396317\pi$
$752$	1.74060	0.0634731
$753$	2.36865	0.0863183
$754$	33.9124	1.23502
$755$	10.8100	0.393417
$756$	0.982703	0.0357406
$757$	53.6251	1.94904	0.974519	−	0.224307i	$-0.0720117\pi$
$757$	53.6251	1.94904	0.974519	+	0.224307i	$0.0720117\pi$
$758$	21.4937	0.780687
$759$	6.16776	0.223876
$760$	4.22917	0.153408
$761$	−42.4729	−1.53964	−0.769821	−	0.638260i	$-0.779654\pi$
$761$	−42.4729	−1.53964	−0.769821	+	0.638260i	$0.779654\pi$
$762$	−0.560962	−0.0203215
$763$	6.23036	0.225554
$764$	20.8836	0.755541
$765$	13.2567	0.479298
$766$	9.28050	0.335318
$767$	12.2950	0.443946
$768$	0.232858	0.00840252
$769$	20.3879	0.735208	0.367604	−	0.929982i	$-0.380178\pi$
$769$	20.3879	0.735208	0.367604	+	0.929982i	$0.380178\pi$
$770$	2.25204	0.0811579
$771$	−2.11862	−0.0763001
$772$	−15.9073	−0.572517
$773$	53.7932	1.93481	0.967404	−	0.253239i	$-0.0814960\pi$
$773$	53.7932	1.93481	0.967404	+	0.253239i	$0.0814960\pi$
$774$	−4.56636	−0.164135
$775$	−0.0712626	−0.00255983
$776$	−14.6988	−0.527658
$777$	1.68529	0.0604593
$778$	−18.7534	−0.672341
$779$	−12.5525	−0.449742
$780$	0.880091	0.0315123
$781$	−7.14539	−0.255682
$782$	37.5682	1.34344
$783$	12.7509	0.455681
$784$	−6.49621	−0.232008
$785$	−3.71542	−0.132609
$786$	0.720212	0.0256891
$787$	−44.0200	−1.56914	−0.784571	−	0.620039i	$-0.787117\pi$
$787$	−44.0200	−1.56914	−0.784571	+	0.620039i	$0.787117\pi$
$788$	14.8401	0.528658
$789$	−0.0876849	−0.00312166
$790$	5.04618	0.179535
$791$	13.8191	0.491349
$792$	−9.10606	−0.323570
$793$	16.8762	0.599291
$794$	16.3237	0.579307
$795$	0.556354	0.0197318
$796$	23.6524	0.838338
$797$	−20.0032	−0.708550	−0.354275	−	0.935141i	$-0.615272\pi$
$797$	−20.0032	−0.708550	−0.354275	+	0.935141i	$0.615272\pi$
$798$	0.680999	0.0241071
$799$	7.63155	0.269985
$800$	−3.94648	−0.139529
$801$	9.08425	0.320976
$802$	−6.39449	−0.225797
$803$	−43.1123	−1.52140
$804$	2.42762	0.0856156
$805$	6.24241	0.220016
$806$	−0.0664917	−0.00234207
$807$	−3.00187	−0.105671
$808$	6.45408	0.227054
$809$	−38.9262	−1.36857	−0.684286	−	0.729214i	$-0.739886\pi$
$809$	−38.9262	−1.36857	−0.684286	+	0.729214i	$0.739886\pi$
$810$	−8.73984	−0.307087
$811$	46.0135	1.61575	0.807877	−	0.589352i	$-0.200617\pi$
$811$	46.0135	1.61575	0.807877	+	0.589352i	$0.200617\pi$
$812$	6.53682	0.229397
$813$	−4.42560	−0.155213
$814$	−31.5203	−1.10479
$815$	21.8224	0.764405
$816$	1.02095	0.0357404
$817$	−6.38709	−0.223456
$818$	−22.7172	−0.794289
$819$	−7.69907	−0.269027
$820$	−3.12695	−0.109198
$821$	5.57537	0.194582	0.0972910	−	0.995256i	$-0.468982\pi$
$821$	5.57537	0.194582	0.0972910	+	0.995256i	$0.468982\pi$
$822$	−3.67505	−0.128182
$823$	−8.66302	−0.301974	−0.150987	−	0.988536i	$-0.548245\pi$
$823$	−8.66302	−0.301974	−0.150987	+	0.988536i	$0.548245\pi$
$824$	−0.330649	−0.0115187
$825$	−2.84073	−0.0989015
$826$	2.36993	0.0824606
$827$	−13.5729	−0.471977	−0.235988	−	0.971756i	$-0.575833\pi$
$827$	−13.5729	−0.471977	−0.235988	+	0.971756i	$0.575833\pi$
$828$	−25.2410	−0.877186
$829$	−34.9131	−1.21258	−0.606290	−	0.795243i	$-0.707343\pi$
$829$	−34.9131	−1.21258	−0.606290	+	0.795243i	$0.707343\pi$
$830$	5.19660	0.180377
$831$	−0.133937	−0.00464622
$832$	−3.68227	−0.127660
$833$	−28.4823	−0.986851
$834$	−2.01652	−0.0698265
$835$	14.7163	0.509280
$836$	−12.7369	−0.440514
$837$	−0.0250006	−0.000864149 0
$838$	20.3370	0.702530
$839$	−34.1485	−1.17894	−0.589469	−	0.807791i	$-0.700663\pi$
$839$	−34.1485	−1.17894	−0.589469	+	0.807791i	$0.700663\pi$
$840$	0.169643	0.00585324
$841$	55.8177	1.92475
$842$	12.5555	0.432690
$843$	3.13132	0.107848
$844$	20.1077	0.692136
$845$	−0.573843	−0.0197408
$846$	−5.12742	−0.176284
$847$	1.02516	0.0352250
$848$	−2.32776	−0.0799357
$849$	−7.27274	−0.249600
$850$	−17.3031	−0.593491
$851$	−87.3709	−2.99504
$852$	−0.538253	−0.0184402
$853$	9.58873	0.328312	0.164156	−	0.986434i	$-0.447510\pi$
$853$	9.58873	0.328312	0.164156	+	0.986434i	$0.447510\pi$
$854$	3.25299	0.111315
$855$	−12.4582	−0.426061
$856$	6.99962	0.239242
$857$	8.85745	0.302565	0.151282	−	0.988491i	$-0.451660\pi$
$857$	8.85745	0.302565	0.151282	+	0.988491i	$0.451660\pi$
$858$	−2.65055	−0.0904882
$859$	−37.4404	−1.27745	−0.638724	−	0.769436i	$-0.720538\pi$
$859$	−37.4404	−1.27745	−0.638724	+	0.769436i	$0.720538\pi$
$860$	−1.59108	−0.0542555
$861$	−0.503515	−0.0171598
$862$	−17.1286	−0.583402
$863$	−18.4029	−0.626440	−0.313220	−	0.949681i	$-0.601408\pi$
$863$	−18.4029	−0.626440	−0.313220	+	0.949681i	$0.601408\pi$
$864$	−1.38452	−0.0471023
$865$	−14.3769	−0.488828
$866$	−31.0475	−1.05504
$867$	0.517713	0.0175825
$868$	−0.0128167	−0.000435027 0
$869$	−15.1974	−0.515538
$870$	2.20118	0.0746270
$871$	−38.3889	−1.30076
$872$	−8.77788	−0.297257
$873$	43.2995	1.46547
$874$	−35.3053	−1.19422
$875$	−6.51775	−0.220340
$876$	−3.24759	−0.109726
$877$	40.1649	1.35627	0.678136	−	0.734937i	$-0.262788\pi$
$877$	40.1649	1.35627	0.678136	+	0.734937i	$0.262788\pi$
$878$	27.5236	0.928878
$879$	−1.12677	−0.0380052
$880$	−3.17287	−0.106958
$881$	−20.6732	−0.696496	−0.348248	−	0.937402i	$-0.613223\pi$
$881$	−20.6732	−0.696496	−0.348248	+	0.937402i	$0.613223\pi$
$882$	19.1364	0.644356
$883$	−35.3486	−1.18958	−0.594788	−	0.803882i	$-0.702764\pi$
$883$	−35.3486	−1.18958	−0.594788	+	0.803882i	$0.702764\pi$
$884$	−16.1447	−0.543004
$885$	0.798042	0.0268259
$886$	−32.4503	−1.09019
$887$	29.6462	0.995423	0.497712	−	0.867343i	$-0.334174\pi$
$887$	29.6462	0.995423	0.497712	+	0.867343i	$0.334174\pi$
$888$	−2.37438	−0.0796790
$889$	1.70988	0.0573476
$890$	3.16528	0.106100
$891$	26.3216	0.881806
$892$	−2.44667	−0.0819205
$893$	−7.17185	−0.239997
$894$	5.53589	0.185148
$895$	−16.5111	−0.551904
$896$	−0.709779	−0.0237121
$897$	−7.34704	−0.245310
$898$	32.0691	1.07016
$899$	−0.166301	−0.00554646
$900$	11.6254	0.387515
$901$	−10.2059	−0.340009
$902$	9.41737	0.313564
$903$	−0.256203	−0.00852590
$904$	−19.4695	−0.647546
$905$	−10.1814	−0.338443
$906$	−2.45242	−0.0814762
$907$	4.69794	0.155993	0.0779963	−	0.996954i	$-0.475148\pi$
$907$	4.69794	0.155993	0.0779963	+	0.996954i	$0.475148\pi$
$908$	9.32125	0.309336
$909$	−19.0123	−0.630598
$910$	−2.68263	−0.0889283
$911$	33.1082	1.09692	0.548462	−	0.836175i	$-0.315213\pi$
$911$	33.1082	1.09692	0.548462	+	0.836175i	$0.315213\pi$
$912$	−0.959452	−0.0317706
$913$	−15.6505	−0.517955
$914$	6.27608	0.207594
$915$	1.09540	0.0362127
$916$	−23.7376	−0.784314
$917$	−2.19530	−0.0724951
$918$	−6.07034	−0.200351
$919$	25.1845	0.830759	0.415379	−	0.909648i	$-0.363649\pi$
$919$	25.1845	0.830759	0.415379	+	0.909648i	$0.363649\pi$
$920$	−8.79487	−0.289958
$921$	0.965003	0.0317979
$922$	−33.7409	−1.11120
$923$	8.51159	0.280163
$924$	−0.510910	−0.0168077
$925$	40.2411	1.32312
$926$	22.6032	0.742787
$927$	0.974018	0.0319910
$928$	−9.20965	−0.302322
$929$	−26.8837	−0.882027	−0.441013	−	0.897500i	$-0.645381\pi$
$929$	−26.8837	−0.882027	−0.441013	+	0.897500i	$0.645381\pi$
$930$	−0.00431584	−0.000141522 0
$931$	26.7666	0.877240
$932$	2.92846	0.0959249
$933$	5.73359	0.187709
$934$	33.0095	1.08010
$935$	−13.9113	−0.454947
$936$	10.8471	0.354550
$937$	8.59670	0.280842	0.140421	−	0.990092i	$-0.455154\pi$
$937$	8.59670	0.280842	0.140421	+	0.990092i	$0.455154\pi$
$938$	−7.39970	−0.241609
$939$	6.68804	0.218256
$940$	−1.78657	−0.0582716
$941$	15.7014	0.511851	0.255925	−	0.966697i	$-0.417620\pi$
$941$	15.7014	0.511851	0.255925	+	0.966697i	$0.417620\pi$
$942$	0.842900	0.0274632
$943$	26.1039	0.850061
$944$	−3.33897	−0.108674
$945$	−1.00866	−0.0328117
$946$	4.79182	0.155796
$947$	18.2670	0.593597	0.296799	−	0.954940i	$-0.404081\pi$
$947$	18.2670	0.593597	0.296799	+	0.954940i	$0.404081\pi$
$948$	−1.14480	−0.0371814
$949$	51.3554	1.66707
$950$	16.2608	0.527570
$951$	4.80531	0.155823
$952$	−3.11198	−0.100860
$953$	25.0302	0.810809	0.405405	−	0.914137i	$-0.367131\pi$
$953$	25.0302	0.810809	0.405405	+	0.914137i	$0.367131\pi$
$954$	6.85707	0.222006
$955$	−21.4352	−0.693626
$956$	3.43686	0.111156
$957$	−6.62924	−0.214293
$958$	−29.5838	−0.955810
$959$	11.2020	0.361732
$960$	−0.239008	−0.00771395
$961$	−30.9997	−0.999989
$962$	37.5470	1.21056
$963$	−20.6193	−0.664449
$964$	−15.8815	−0.511509
$965$	16.3275	0.525600
$966$	−1.41619	−0.0455651
$967$	−46.4273	−1.49300	−0.746500	−	0.665385i	$-0.768267\pi$
$967$	−46.4273	−1.49300	−0.746500	+	0.665385i	$0.768267\pi$
$968$	−1.44434	−0.0464228
$969$	−4.20666	−0.135137
$970$	15.0871	0.484417
$971$	−57.4737	−1.84442	−0.922209	−	0.386692i	$-0.873618\pi$
$971$	−57.4737	−1.84442	−0.922209	+	0.386692i	$0.873618\pi$
$972$	6.13632	0.196823
$973$	6.14662	0.197052
$974$	−5.88070	−0.188430
$975$	3.38388	0.108371
$976$	−4.58310	−0.146701
$977$	48.0063	1.53586	0.767929	−	0.640535i	$-0.221287\pi$
$977$	48.0063	1.53586	0.767929	+	0.640535i	$0.221287\pi$
$978$	−4.95074	−0.158307
$979$	−9.53278	−0.304669
$980$	6.66780	0.212995
$981$	25.8577	0.825572
$982$	−21.1468	−0.674822
$983$	57.9667	1.84885	0.924426	−	0.381362i	$-0.124545\pi$
$983$	57.9667	1.84885	0.924426	+	0.381362i	$0.124545\pi$
$984$	0.709397	0.0226147
$985$	−15.2321	−0.485335
$986$	−40.3792	−1.28593
$987$	−0.287682	−0.00915701
$988$	15.1722	0.482691
$989$	13.2824	0.422357
$990$	9.34658	0.297054
$991$	−8.14907	−0.258864	−0.129432	−	0.991588i	$-0.541315\pi$
$991$	−8.14907	−0.258864	−0.129432	+	0.991588i	$0.541315\pi$
$992$	0.0180573	0.000573319 0
$993$	5.78667	0.183635
$994$	1.64066	0.0520387
$995$	−24.2772	−0.769638
$996$	−1.17893	−0.0373558
$997$	−48.0425	−1.52152	−0.760761	−	0.649032i	$-0.775174\pi$
$997$	−48.0425	−1.52152	−0.760761	+	0.649032i	$0.775174\pi$
$998$	−13.9679	−0.442145
$999$	14.1175	0.446659

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.f.1.20	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.f.1.20	✓	31	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.232858 q^{3} +1.00000 q^{4} -1.02641 q^{5} +0.232858 q^{6} -0.709779 q^{7} +1.00000 q^{8} -2.94578 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.232858 q^{3} +1.00000 q^{4} -1.02641 q^{5} +0.232858 q^{6} -0.709779 q^{7} +1.00000 q^{8} -2.94578 q^{9} -1.02641 q^{10} +3.09122 q^{11} +0.232858 q^{12} -3.68227 q^{13} -0.709779 q^{14} -0.239008 q^{15} +1.00000 q^{16} +4.38444 q^{17} -2.94578 q^{18} -4.12034 q^{19} -1.02641 q^{20} -0.165277 q^{21} +3.09122 q^{22} +8.56854 q^{23} +0.232858 q^{24} -3.94648 q^{25} -3.68227 q^{26} -1.38452 q^{27} -0.709779 q^{28} -9.20965 q^{29} -0.239008 q^{30} +0.0180573 q^{31} +1.00000 q^{32} +0.719815 q^{33} +4.38444 q^{34} +0.728527 q^{35} -2.94578 q^{36} -10.1967 q^{37} -4.12034 q^{38} -0.857443 q^{39} -1.02641 q^{40} +3.04649 q^{41} -0.165277 q^{42} +1.55014 q^{43} +3.09122 q^{44} +3.02359 q^{45} +8.56854 q^{46} +1.74060 q^{47} +0.232858 q^{48} -6.49621 q^{49} -3.94648 q^{50} +1.02095 q^{51} -3.68227 q^{52} -2.32776 q^{53} -1.38452 q^{54} -3.17287 q^{55} -0.709779 q^{56} -0.959452 q^{57} -9.20965 q^{58} -3.33897 q^{59} -0.239008 q^{60} -4.58310 q^{61} +0.0180573 q^{62} +2.09085 q^{63} +1.00000 q^{64} +3.77953 q^{65} +0.719815 q^{66} +10.4253 q^{67} +4.38444 q^{68} +1.99525 q^{69} +0.728527 q^{70} -2.31151 q^{71} -2.94578 q^{72} -13.9467 q^{73} -10.1967 q^{74} -0.918966 q^{75} -4.12034 q^{76} -2.19409 q^{77} -0.857443 q^{78} -4.91632 q^{79} -1.02641 q^{80} +8.51494 q^{81} +3.04649 q^{82} -5.06288 q^{83} -0.165277 q^{84} -4.50025 q^{85} +1.55014 q^{86} -2.14454 q^{87} +3.09122 q^{88} -3.08382 q^{89} +3.02359 q^{90} +2.61360 q^{91} +8.56854 q^{92} +0.00420477 q^{93} +1.74060 q^{94} +4.22917 q^{95} +0.232858 q^{96} -14.6988 q^{97} -6.49621 q^{98} -9.10606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9	\( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9} - 23 q^{10} - 32 q^{11} - 13 q^{12} - 8 q^{13} - 18 q^{14} - 14 q^{15} + 31 q^{16} - 30 q^{17} + 20 q^{18} - 38 q^{19} - 23 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 13 q^{24} + 40 q^{25} - 8 q^{26} - 28 q^{27} - 18 q^{28} - 7 q^{29} - 14 q^{30} - 32 q^{31} + 31 q^{32} - 9 q^{33} - 30 q^{34} - 14 q^{35} + 20 q^{36} + 11 q^{37} - 38 q^{38} - 9 q^{39} - 23 q^{40} - 76 q^{41} - 16 q^{42} - 33 q^{43} - 32 q^{44} - 40 q^{45} - 24 q^{46} - 96 q^{47} - 13 q^{48} + 15 q^{49} + 40 q^{50} - 55 q^{51} - 8 q^{52} - 28 q^{53} - 28 q^{54} - 52 q^{55} - 18 q^{56} - 21 q^{57} - 7 q^{58} - 72 q^{59} - 14 q^{60} - 9 q^{61} - 32 q^{62} - 54 q^{63} + 31 q^{64} - 38 q^{65} - 9 q^{66} - 4 q^{67} - 30 q^{68} - 17 q^{69} - 14 q^{70} - 61 q^{71} + 20 q^{72} - 62 q^{73} + 11 q^{74} - 63 q^{75} - 38 q^{76} - 9 q^{77} - 9 q^{78} - 30 q^{79} - 23 q^{80} - 13 q^{81} - 76 q^{82} - 90 q^{83} - 16 q^{84} + 26 q^{85} - 33 q^{86} - 34 q^{87} - 32 q^{88} - 99 q^{89} - 40 q^{90} - 47 q^{91} - 24 q^{92} - 6 q^{93} - 96 q^{94} - 24 q^{95} - 13 q^{96} - 46 q^{97} + 15 q^{98} - 48 q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 13 * q^3 + 31 * q^4 - 23 * q^5 - 13 * q^6 - 18 * q^7 + 31 * q^8 + 20 * q^9 - 23 * q^10 - 32 * q^11 - 13 * q^12 - 8 * q^13 - 18 * q^14 - 14 * q^15 + 31 * q^16 - 30 * q^17 + 20 * q^18 - 38 * q^19 - 23 * q^20 - 16 * q^21 - 32 * q^22 - 24 * q^23 - 13 * q^24 + 40 * q^25 - 8 * q^26 - 28 * q^27 - 18 * q^28 - 7 * q^29 - 14 * q^30 - 32 * q^31 + 31 * q^32 - 9 * q^33 - 30 * q^34 - 14 * q^35 + 20 * q^36 + 11 * q^37 - 38 * q^38 - 9 * q^39 - 23 * q^40 - 76 * q^41 - 16 * q^42 - 33 * q^43 - 32 * q^44 - 40 * q^45 - 24 * q^46 - 96 * q^47 - 13 * q^48 + 15 * q^49 + 40 * q^50 - 55 * q^51 - 8 * q^52 - 28 * q^53 - 28 * q^54 - 52 * q^55 - 18 * q^56 - 21 * q^57 - 7 * q^58 - 72 * q^59 - 14 * q^60 - 9 * q^61 - 32 * q^62 - 54 * q^63 + 31 * q^64 - 38 * q^65 - 9 * q^66 - 4 * q^67 - 30 * q^68 - 17 * q^69 - 14 * q^70 - 61 * q^71 + 20 * q^72 - 62 * q^73 + 11 * q^74 - 63 * q^75 - 38 * q^76 - 9 * q^77 - 9 * q^78 - 30 * q^79 - 23 * q^80 - 13 * q^81 - 76 * q^82 - 90 * q^83 - 16 * q^84 + 26 * q^85 - 33 * q^86 - 34 * q^87 - 32 * q^88 - 99 * q^89 - 40 * q^90 - 47 * q^91 - 24 * q^92 - 6 * q^93 - 96 * q^94 - 24 * q^95 - 13 * q^96 - 46 * q^97 + 15 * q^98 - 48 * q^99

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